This preview shows page 1. Sign up to view the full content.
Unformatted text preview: Q . (ii) Prove that I is the only element in Q of order 2, and that all other elements M = I satisfy M 2 = I . Solution. Straightforward multiplication. (iii) Show that Q has a unique subgroup of order 2, and it is the center of Q . Solution. A group of order 2 must be a cyclic group generated by an element of order 2. It is shown, in part (i), that I is the only element of order 2. It is clear that h I i Z ( Q ) , for scalar matrices commute with every matrix. On the other hand, for every element M = I , there is N Q with M N = N M . (iv) Prove that h I i is the center Z ( Q ) . Solution. For each M Q but not in h I i , there is a matrix M Q with M M = M M . 2.87 Prove that the quaternions Q and the dihedral group D 8 are nonisomorphic groups of order 8....
View Full Document
- Fall '11